Find a formula for the intersection math

Let I=(a,b) and J=(c,d) with I and J having a nonempty intersection. Find a formula for the intersection of I and J and prove it.

When a<c I have found that the intersection is [c,b]. Now I need to prove that it is. The way I intend to prove it is by showing that [c,b] and the intersection of I and J are both subsets of each other. I've done one, [c,b] being a subset of I intersect J.

I intersect J being a subset of [c,b] is a bit trickier. I've broken it down to:

let x be an element of I intersect J then x is an element of I and x is an element of J. Then:

Let I=(a,b) and J=(c,d) with I and J having a nonempty intersection. Find a formula for the intersection of I and J and prove it.

When a<c I have found that the intersection is [c,b]. Now I need to prove that it is. The way I intend to prove it is by showing that [c,b] and the intersection of I and J are both subsets of each other. I've done one, [c,b] being a subset of I intersect J.

I intersect J being a subset of [c,b] is a bit trickier. I've broken it down to:

let x be an element of I intersect J then x is an element of I and x is an element of J. Then:

If you want to do well in Analysis, you are going to have to make an effort to at least write things correctly!. (a, b) is not the same as [a,b] and you are confusing the two.

Jason Rox said:

When a<c I have found that the intersection is [c,b].

No, you haven't- for two reasons. First, as others said, the intersection of the two open intervals, (a,b) and (c,d) cannot be the closed interval (c,b)- you are being careless with your notation. More importantly, what if a= 0, b= 10, c= 1, d= 2- that is, I= (0, 10) and J= (1, 2). Then the intersection is (1, 2), not (1, 10).

If you want to do well in Analysis, you are going to have to make an effort to at least write things correctly!. (a, b) is not the same as [a,b] and you are confusing the two.

No, you haven't- for two reasons. First, as others said, the intersection of the two open intervals, (a,b) and (c,d) cannot be the closed interval (c,b)- you are being careless with your notation. More importantly, what if a= 0, b= 10, c= 1, d= 2- that is, I= (0, 10) and J= (1, 2). Then the intersection is (1, 2), not (1, 10).

I wasn't being careless in notation. I do know the difference between closed and open intervals, and their specific notation. I did indeed use notation for both open and closed intervals in my OP. I made the mistake that I thought the intersection would be a closed interval (which is why I used closed interval notation)...since they overlap they would have both, but I realize the error in this. I hope you realize your error too.